1. Prior Art
My above two patents disclose a method for transforming a globe represented in a spherical, three-dimensional round format into a flat disc representation using an Azimuthal Equidistant (AE) projection format.
The AE projection is an azimuthal map projection which preserves azimuths (angles or directions) and roundness in general. The projection shows places on the globe in an equidistant (at equal distances) manner, specifically it shows distances from a central point on the map to any other point in an equally spacing. Thus a great circle around Earth can be drawn as a straight line, contrary to wavy lines drawn in the Mercator or other projections. A great circle is a circle with the same diameter as the Earth where any given two points on the circle constitute the shortest distance between these points on the Earth's surface. As great circle lines are always straight on the real Earth when viewed from overhead, the wavy lines on most maps do not reflect reality.
An ordinary world map drawn in the AE projection has serious distortion problem at its periphery. This is because this map requires that all configurations be drawn within one large circle. To overcome this distortion problem, in my previous maps I divided this one large circle into two small circles with half the diameter of the large circle. One of the small circles with any given center point of interest is a main hemisphere, which represents inner half circle of the large AE projection map. Another small circle having the same-size is an opposing hemisphere centered on an antipodal point (a point directly or exactly opposite a given point on Earth), which represents an outer, doughnut-shaped area of the large AE projection map. To distinguish the latter projection with two hemispheres from the ordinary AE projection, I called it an Azimuthal Equidistant Hemispheric projection.
My above '790 patent shows the opposing hemisphere divided in half to provide two resulting semi-hemispheres attached to the right and left sides of the main hemisphere. This procedure enables the continental hemisphere to be featured prominently.
My '575 patent provides a world map projection in which any place in the world can be the center of interest. Also in this projection, the opposing hemisphere is also divided in half, using the same representation as in my '790 patent.
In addition to my own patents, U.S. Pat. No. 5,902,113 to Paul L. Pryor et al., May 11, 1999, entitled, “Map and Calculator Device” shows a map drawn in two hemispheres and in the AE projection. However it differs from the maps of my previous patents in one important aspect: All of my maps show the world in one view. On the other hand, Pryor et al. locate one half of the world on an obverse side of a sheet and the other half on the reverse side. In other words, the two hemispheres are back to back. A user of this map must flip again and again to view any relation of two points separated between obverse and reverse sides.
The Pryor et al. map has elements which can locate antipodal points of any given center point along a great circle line. If the one of elements is moved from the center point, the other of elements is moved from the antipodal point accordingly. Using the same hemispheric map with the same center point, one can still determine the antipodal point of any given point. If one can move the center point and its antipodal point freely, one need not limit the starting point of measurement to the center point. Rather the distance from any point to any other point can be measured correctly if they are located on a line along a great circle.
A poster entitled “Map Projections” by the US Geological Survey, states “Distances and directions to all places [are] true only from center point of projection. Distances [are] correct between points along straight lines through center. All other distances [are] incorrect”. This statement is correct but is likely to mislead. The importance of a great circle is that distances between any two points are correct, not just where one point is the center point of projection.
The same thing, however, does not apply to directions. For example, Florida is located due east of San Francisco, but reversely, San Francisco is not located due west of Florida. Directions from all places along a great circle are not true, except from the center point and from its antipode. It is because the Earth is round and north-south lines that determine the bearings are not parallel. I will explain this in more detail under Detailed Description below.
In my '790 patent, FIG. 29 relates to another idea regarding the AE projection, namely the embodiment, “Two Bowl-Shaped Hemispheric Maps”. This has merit since it is the only projection which is directly transferable to a globe.
However, its transferability has not been explored fully. Besides, my prior AE Hemispheric projection requires that the opposing hemisphere be divided and it requires means for moving it around the border of other hemisphere in a coordinated way using intermeshed gears.
2. Advantages
The present map integrates further ideas and developments to provide improved embodiments. Accordingly, several advantages of one or more aspects of the present embodiments are:                1. A world map can show how directions of a center point and its antipodal point differ from all other points along a great circle.        2. A world map in the AE projection can be used to measure distances between any two points along a great circle.        3. A world map in the AE Hemispheric projection can show how easily it can be transformed into a spherical form.        4. A hemispheric world map with any center point can be acquired instantly.        
Still further advantages of various embodiments will become apparent from consideration of the ensuing description and drawings.